TY - JOUR

T1 - Many T copies in H-free graphs

AU - Alon, Noga

AU - Shikhelman, Clara

N1 - Publisher Copyright:
© 2016 Elsevier Inc.

PY - 2016/11/1

Y1 - 2016/11/1

N2 - For two graphs T and H with no isolated vertices and for an integer n, let ex(n,T,H) denote the maximum possible number of copies of T in an H-free graph on n vertices. The study of this function when T=K2 is a single edge is the main subject of extremal graph theory. In the present paper we investigate the general function, focusing on the cases of triangles, complete graphs, complete bipartite graphs and trees. These cases reveal several interesting phenomena. Three representative results are: (i) ex(n,K3,C5)≤(1+o(1))32n3/2,(ii) For any fixed m, s≥2m−2 and t≥(s−1)!+1, ex(n,Km,Ks,t)=Θ(nm−(m2)/s), and(iii) For any two trees H and T, ex(n,T,H)=Θ(nm) where m=m(T,H) is an integer depending on H and T (its precise definition is given in Section 1).The first result improves (slightly) an estimate of Bollobás and Győri. The proofs combine combinatorial and probabilistic arguments with simple spectral techniques.

AB - For two graphs T and H with no isolated vertices and for an integer n, let ex(n,T,H) denote the maximum possible number of copies of T in an H-free graph on n vertices. The study of this function when T=K2 is a single edge is the main subject of extremal graph theory. In the present paper we investigate the general function, focusing on the cases of triangles, complete graphs, complete bipartite graphs and trees. These cases reveal several interesting phenomena. Three representative results are: (i) ex(n,K3,C5)≤(1+o(1))32n3/2,(ii) For any fixed m, s≥2m−2 and t≥(s−1)!+1, ex(n,Km,Ks,t)=Θ(nm−(m2)/s), and(iii) For any two trees H and T, ex(n,T,H)=Θ(nm) where m=m(T,H) is an integer depending on H and T (its precise definition is given in Section 1).The first result improves (slightly) an estimate of Bollobás and Győri. The proofs combine combinatorial and probabilistic arguments with simple spectral techniques.

KW - Complete bipartite graphs

KW - Complete graphs

KW - Extremal graph theory

KW - H-free graphs

KW - Projective norm graphs

UR - http://www.scopus.com/inward/record.url?scp=84962689519&partnerID=8YFLogxK

U2 - 10.1016/j.jctb.2016.03.004

DO - 10.1016/j.jctb.2016.03.004

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AN - SCOPUS:84962689519

SN - 0095-8956

VL - 121

SP - 146

EP - 172

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

ER -